Trapezoidal Fuzzy Numbers Research Articles

Analysing skewness of credibilistic coherent trapezoidal fuzzy numbers: Implications for portfolio selection

Analysing skewness of credibilistic coherent trapezoidal fuzzy numbers: Implications for portfolio selection

Bounded Closed Interval-Valued Decagonal Fuzzy Number and its Application

In this paper, we introduced the notation of Algebraic operations of bounded closed interval-valued decagonal fuzzy number. To overcome the uncertainty here we used fuzzy numbers. Many researchers have focused their research with Triangular and Trapezoidal fuzzy numbers, in some cases it is not fit to the problem when the vagueness arises in ten different opinions so our motto is to develop the theorems for bounded closed interval using Algebraic operations such as Addition, Subtraction, Multiplication and Division via Decagonal Fuzzy Number.

Fermatean trapezoidal fuzziness average aggregation scheme for selection of infant clothing by group decision-making

The textile industry has gradually increased its use of various chemicals and the quantity of residues produced. Newborn baby apparel should be made of safe materials since it often comes into touch with them. Infant cloth selection is determined via group decision-making in Fermatean trapezoidal fuzzy number (FTFN). This study provides a new score and accuracy function and weighted, ordered, and hybrid averaging aggregation operators on FTFN. The properties of the proposed averaging aggregation operators on FTFN are presented. An algorithm for the multi-attribute group decision-making (MAGDM) approach will be proposed by applying the proposed averaging aggregation operators. The case studies concentrate on choosing newborn clothing with reduced chemical content within the textile sector. The presentation of the sensitivity analysis of the criterion weights is conducted to guarantee the durability and stability of the framework that has been implemented. Furthermore, a comprehensive analysis compares the novel framework with previous models, highlighting its superior performance.

A Novel Method Based on the Fuzzy Entropy Measure to Optimize the Fuzziness in Trapezoidal Strong Fuzzy Partitions

Analyzing the uncertainty of outcomes based on estimates of the data’s membership degrees to fuzzy sets is essential for making decisions. These fuzzy sets are often designated by experts as strong fuzzy partitions of the data domain with trapezoidal fuzzy numbers. Some indices of the fuzzy set’s fuzziness provide an assessment of the degree of uncertainty of the results. It is feasible to bring the fuzzy sets’ fuzziness below a tolerable level by suitably redefining the strong fuzzy partition. Significant differences in the original fuzzy partition, however, result in disparities concerning the decision maker’s approximative reasoning and the interpretability of the results. In light of this, we provide in this study a technique applied to trapezoidal strong fuzzy partitions that, while not appreciably altering the original fuzzy partition, reduces the fuzziness of its fuzzy sets. The fuzziness of the fuzzy sets is assessed using the De Luca and Termini fuzzy entropy. An iterative process is then executed, with the aim of modifying the cores of the trapezoidal fuzzy partitions to decrease their fuzziness. This technique is tested on datasets containing average daily temperatures measured in various cities. The findings demonstrate that this approach strikes a great balance between the goal of lessening the fuzziness of the fuzzy sets and the goal of not appreciably altering the original fuzzy partition.

Enhancing group decision-making: Maximum consensus aggregation for fuzzy cross-efficiency under hesitant fuzzy linguistic information

Group decision-making (GDM) is essential as it recognizes the inherent complexity of many decision scenarios, which frequently require the collective wisdom and knowledge of multiple decision-makers (DMs) to be effectively resolved. The proposed method aims to develop fuzzy data envelopment analysis (DEA) cross-efficiency models tailored to address GDM challenges, wherein attribute values are provided by DMs using hesitant fuzzy linguistic term sets (HFLTSs). For this purpose, we initially transform HFLTSs into their corresponding fuzzy envelopes, defined as trapezoidal fuzzy numbers (TrFNs). This conversion strategy effectively minimizes the loss in assessments based on HFLTSs while retaining the inherent ambiguity of the original information. Building upon this foundation, we develop fuzzy cross-efficiency models by leveraging the α-level sets of fuzzy envelopes. These models are designed to handle fuzzy input and output variables under various α-level sets, which are capable of considering all possible attribute values for each alternative. Following this, we implement a maximum consensus model using fuzzy cross-efficiency to assign weights to DMs. These weights facilitate the aggregation of individual fuzzy cross-efficiency intervals obtained from DMs’ assessments into collective ones, which serve to rank alternatives. Finally, we showcase the effectiveness and superiority of our proposal through numerical validation and comparative analysis.

Linear fractional transportation problem in bipolar fuzzy environment

This article proposes a solution methodology for the Linear Fractional Transportation Problem (LFTP) by incorporating bipolar fuzzy sets (BFSs) to accommodate both positive and negative judgmental perspectives. The approach explores Zimmermann's extension within bipolar fuzzy environment to compare outcomes. In this context, the cost function and constraint coefficients are depicted using interval-valued trapezoidal bipolar fuzzy numbers (IVTrBFNs) and defuzzified by (s, t)-cut. The initial approach employs the simplex method and fuzzy optimization method, renowned for its effectiveness in obtaining optimal solutions. In the alternative method, Bipolar fuzzy programming approach (BFPA) is utilized for better outcome. In this method the LFTP is altered to a Multi-Objective Transportation Problem (MOTP), and BFPA extends Zimmermann's technique under suitable positive and negative membership functions, converting MOTP to a one-objective transportation problem (TP) and solved using LINGO software. Supporting this proposed method some theorems are formulated to demonstrate that the most effective solution of the single-objective TP is a Pareto optimal solution for the corresponding MOTP. A quantitative example is provided for better understanding of the proposed BFPA method alongside two other approaches.

Numerical solution of fuzzy non-linear equations under generalized trapezoidal fuzziness by Newton–Raphson method

In the real world, we are faced with many problems that cannot be easily described by crisp values, as there is always some uncertainty involved in real-life situations and in those situations, we end up receiving fuzzy values rather than crisp ones in those situations. Some preliminary concepts of fuzzy sets and operations on trapezoidal fuzzy numbers incorporated with the membership function have been discussed here. This paper aims to demonstrate the advantages of the newly developed fuzzified newton-raphson method with the help of trapezoidal fuzzy numbers for solving algebraic nonlinear equations arising in fuzzy environments. The algorithm of newton-raphson techniques for solving the same has been described. This paper presents several examples of how the new fuzzified newton-raphson method works.

Двоетапний метод розв’язання задачі комівояжера на основі генетичного алгоритму

One of the main tasks in logistics is to find the most efficient route in the traveling salesman problem on a given transportation network, allowing for the servicing of the maximum number of customers while considering certain criteria. In the typical traveling salesman problem, the objective function is most often the route length or duration. However, such a formulation does not account for the subjectivity in evaluating the travel durations at different stages, which may be influenced by various objective and subjective factors. The problem of the traveling salesman with fuzzily defined travel durations between network nodes is considered. The approach to solving the problem is based on a genetic algorithm with enhanced mutation procedures and population diversity generation. Trapezoidal fuzzy numbers are used to formalize fuzzy values, presented in the generalized case based on the application of Gaussian distribution and corresponding characteristics. A two-stage scheme for solving the optimization problem of the traveling salesman on a given transportation network is proposed, combining the genetic algorithm and Ward's clustering method. In the first stage, a clustering procedure is conducted using Ward's method. In the second stage, based on the obtained set of clusters and the identified optimal inter-cluster distances, route duration optimization is performed within the clusters. To test the efficiency of the developed method, two types of models were generated: one with a limited number of available paths and one with a fully connected topology. Computational experiments were conducted to apply the developed method to solving the traveling salesman problem with travel durations between cities represented as fuzzy numbers. A comparison of numerical calculation results was performed. The obtained results demonstrated the effectiveness of preliminary clustering when using the genetic algorithm to find the best local optima under the same input parameters.

Causal Analysis of Roof Caving on Underground Mine: A New Theory and Optimized DEMATEL Approach

In the context of mines, roof-caving incidents constitute the most common and expensive accidents. To enhance the management and prevention of roof-caving accidents, it is imperative to investigate the factors that contribute to such incidents and comprehend the intricate causal relationships among them. This study aims to classify the causes of these accidents into three categories: basic factors, controllable factors, and sudden factors, based on the mechanism of roof caving. The categorization is primarily determined by two indicators: intervisibility and variability. Furthermore, the study delves into analyzing the mutual influence relationships among these factors and proposes the BCX theory (Basic-Controllable-Sudden causing theory) for roof caving. Subsequently, based on this theory, an index system called BCX is established for roof caving, and the DEMATEL method is employed to analyze the factors within this index system. To attain more accurate results, this study utilizes interval trapezoidal type-2 fuzzy number scale optimization and Tsallis relative entropy to address the limitations of the DEMATEL method. By comparing the outcomes of the traditional and optimal DEMATEL methods, it is observed that the optimal method exhibits superior applicability in the BCX index system of roof caving, with results that align closely with the actual scenario. Therefore, the optimal DEMATEL method’s analysis of centrality, importance, and chain relationships between the factors within the BCX index system will offer valuable guidance for preventing roof-caving accidents in mining operations.

Dynamic ranking function to optimize transshipment costs in intuitionistic Type-2 and Type-1 fuzzy environments

In the dynamic realm of organizational logistics, accurately minimizing transportation and transshipment costs is crucial, yet often challenging due to inherent uncertainties. This paper introduces a novel application of fuzzy logic to provide a more precise analysis of these costs. Specifically, it develops an innovative ranking function for trapezoidal fuzzy numbers (TrFNs) for Type-2 and Type-1 fuzzy environments, a tool yet unexplored in existing literature. The main contributions of this paper are the idea that a ranking function for TrFNs can significantly improve decision-maker's freedom in cost analysis due to an adherence on all (a, b, c d) parameters of TrFN. A new decision-oriented ranking method for these fuzzy numbers is developed which consists of an inventive algorithm. The method is also considered for intuitionistic TrFNs and applied to solve transshipment costs in fuzzy area. To verify the proposed methodology's efficiency, effectiveness and accuracy a numerical example in Wolfram Mathematica 9.0 is demonstrated showing superior computational performance over existing methods.

Selection of Military Armored Vehicle Using Fuzzy EDAS method

Military armored vehicles are specially designed and equipped for defense and security activities. During military operations, the right vehicle selection is of vital importance for the success of the operation. This problem examined 5 different alternative military armored vehicles for operational and logistic success using the criteria Engine Power, Dimensions, Fire Power Range, Gradeability, and Armor Thickness. In military decision problems, some alternatives probably need to be evaluated regarding more than one conflicting criterion. The decision among these alternatives will usually involve other strategic-level decisions. Multi-criteria decision-making (MCDM) refers to making decisions in such a situation. Are there any methods and techniques available to solve MCDM problems? The evaluation according to distance from average solution (EDAS) method is a multi-criteria decision-making method whose effectiveness has been tested in other problem areas in the literature. In this study, we use the EDAS method to address the problem of military armored vehicle selection in a fuzzy environment. While a Type-1 fuzzy set is used in the fuzzy EDAS method, trapezoidal fuzzy numbers are preferred for military expert evaluation. The results of this study show that the fuzzy EDAS method is effective.

Fuzzy uncertainty modeling of generalized Hirota–Satsuma coupled Korteweg–de Vries equation

This article explores the solitary wave solutions of a generalized Hirota–Satsuma Coupled Korteweg–de Vries (HSCKdV) equation. The HSCKdV equation is a mathematical model that describes certain types of long waves, particularly those found in shallow water. The generalized HSCKdV equation is solved exactly using the Homotopy Perturbation Transform Method (HPTM). By applying this technique, the authors obtain solutions in the form of a convergent power series. These solutions offer an understanding of the characteristics of solitary waves within the domain of shallow water waves. The HSCKdV equation has been solved using the adomian decomposition method, and the results have been compared with those obtained from the HPTM. This comparison demonstrates the effectiveness of the HPTM in solving such nonlinear equations. Further, the HSCKdV equation is extended to a fuzzy version considering the initial condition as a fuzzy parameter. Uncertainty in the initial condition is addressed by representing it using triangular and trapezoidal fuzzy numbers. The generalized fuzzy HSCKdV equation is subsequently tackled using the fuzzy HPTM (FHPTM) providing fuzzy bound solutions. Using the FHPTM, we explain the fuzzy results, highlighting how the solitary wave splits into two solitary waves and noting that the lower and upper bound solutions are interchanged due to negative fuzzy results.

Expected value of generalized trapezoidal bipolar fuzzy number to solve a multi-item marketing planning inventory model with allowable shortages

Decision makers often grapple with the complexity of multi-faceted real-life problems, where uncertainty arises from both positive and negative states of mind. This duality reflects the decision maker’s optimistic and pessimistic perspectives during the decision-making process. To address this, our paper introduces innovative concepts and frameworks for generalized bipolar trapezoidal fuzzy numbers.We propose novel arithmetic operations on these fuzzy numbers, computing the negative α-cut and positive β-cut methods. Furthermore, we uniquely compute the convex combination of expected values from both the positive and negative membership parts.These theoretical advancements are applied to a practical case study: a multi-item marketing planning inventory model with allowable shortages. Our proposed method’s efficacy is highlighted through detailed numerical illustrations, sensitivity analyses and comparative studies complemented by compelling graphical presentations

Optimization techniques of Assignment Problem using Trapezoidal Intuitionistic Fuzzy Numbers and Interval Arithmetic

This paper discusses the Assignment problem to optimize the assigning of jobs to workers based on their talents and efficiency. In general, scheduling jobs plays a significant role in manufacturing and is advantageous in real world applications as we face more uncertainty and ambiguity in assigning jobs. The Intuitionistic Fuzzy Assignment problem (IFAP) is employed in circumstances when decision-makers have to deal with uncertainty. The domains are Trapezoidal Intuitionistic Fuzzy Numbers (TrIFNs) and the techniques used are Hungarian Method (HM), Brute Force Method (BFM), and Greedy Method (GM). The suggested model's performance is compared with the existing approach with the help of interval arithmetic operations. Allocating work to the individual is illustrated numerically, the optimal solution of minimizing cost is obtained using R programming and the results of comparative analysis are shown diagrammatically that help viewers to easily understand and generate results from comparisons.

Stability Analysis of Diabetes Mellitus Model in Neutrosophic Fuzzy Environment

Neutrosophic differential equation (NDE) plays a vital role in mathematical modeling on uncertainty during last few years.It is established that NDE comprises a neutrosophic number whose membership function contains three parts like as truth function, indeterministic and falsity functions.In this paper, a diabetes mellitus model has been formulated in neutrosophic fuzzy environment to get more realistic mathematical model. The initial conditions of diabetes system are considered as trapezoidal neutrosophic fuzzy number. The concept of NDE via nuetrosophic generalised derivative of type-1 and type-2 has used in this article. This is the first article in which the stability analysis of diabetes model in neutrosophic environment is introduced. The existence of strong solution and weak neutrosophic solution of the proposed system are employed in the manuscript. Also, we have exposed the comparison of the suggested model between fuzzy environment and neutrosophic environment.The all results, formulas are confirmed graphically and numerically by MATLAB software.

A Decision-Making Framework for Sustainable Highway Restaurant Site Selection: AHP-TOPSIS Approach based on the Fuzzy Numbers

This research work indicates the site selection problem for a restaurant beside the highway. It is really challenging as well as demanding work. It mainly depends on the cost of the selected location, transportation, parking facility, hygienic atmosphere, distance from the city, etc. In this paper, the proposed site selection problem is constructed and solved by using a fuzzy MUlti-Criteria Decision Making (MCDM) process. The two methods of MCDM, that we use here namely the Analytic Hierarchy Process (AHP) and the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) for calculate criteria weight and rank the proposed restaurant locations, respectively. Multiple decision makers (DMs) opinions are considered for data collection and Trapezoidal Fuzzy Numbers (TrFNs) are taken as uncertainty tools to deal with uncertainty. Finally, we perform the sensitivity and comparative analysis performed to verify the system's vagueness and flexibility.

A Malmquist fuzzy data envelopment analysis model for performance evaluation of rural healthcare systems

The primary purpose of this article is to measure the relative efficiency and productivity change over time in rural healthcare systems in the presence of fuzzy data. First, a novel ranking function based on the lower and upper bounds of alpha-cut of the trapezoidal fuzzy numbers (TrFNs) is proposed to compare the TrFNs. The suggested ranking technique is used to construct the fuzzy data envelopment analysis (FDEA), Malmquist fuzzy DEA (Mal-FDEA), and undesirable Malmquist fuzzy DEA (UN-Mal-FDEA ) models. The proposed models evaluate the efficiency and productivity of decision-making units (DMUs) when the input and output data are given in the form of TrFNs. In addition, a case study of the rural healthcare system in a developing country has been considered to demonstrate the applicability of the developed models. The work considers number of sub-centers (SCs), the number of primary health centers (PHCs), the number of community health centers (CHCs), nursing Staff at PHCs, an auxiliary nurse and midwives (ANM) at SCs, doctors at PHCs, pharmacists at PHCs, laboratory technicians at PHCs, radiographers at CHCs, and specialists at CHCs as input parameters and average population covered by CHCs, average village covered by CHCs, number of patients, and infant mortality rates as output parameters to analyze the performance of the rural healthcare systems. We show the UN-Mal-FDEA model has a higher production value than the Mal-FDEA model. The results of our proposed models enable us to recognize inefficiencies that states may rectify without compromising healthcare quality.

Assessing the priority levels of motivational factors for lecturers: A case study at vietnam national university, Hanoi

Creating motivation plays a crucial role in enhancing the job performance of university lecturers. This study aims to assess the motivational factors among lecturers at Vietnam National University, Hanoi. Five factors with 30 sub-factors influencing lecturers' motivation are considered in this study, including: salary and benefits (S_B, including 6 sub-factors), leadership skills (L_S, including 7 sub-factors), training and professional development opportunities (T_D, including 6 sub-factors), working conditions and environment (W_C, including 5 sub-factors), and recognition and rewards for accomplishments (R_R, including 6 sub-factors). This study applies the Analytic Hierarchy Process (AHP) method using a generalized fuzzy set for analysis. The steps of the fuzzy Analytic Hierarchy Process include: determining the comparison matrix of generalized trapezoidal fuzzy numbers, calculating the values of the fuzzy synthetic extents, and defining the weight vector of generalized trapezoidal fuzzy numbers. Data for the study were collected through expert interviews employing a questionnaire. The analysis results indicate that leadership skills and salary and benefits are the most influential factors on lecturers' motivation at Vietnam National University, Hanoi, followed by working conditions and environment, recognition and rewards for accomplishments, and training and professional development opportunities. Based on the analysis results, the study proposes several solutions to create motivation among lecturers at Vietnam National University, Hanoi, including: VNU-Hanoi and its affiliated units should prioritize bolstering the training of lecturer teams, particularly focusing on educational and training management staff. Furthermore, additional resources should be allocated to improve lecturer income, welfare, and rewards, alongside increasing investment and optimizing the utilization of infrastructure and educational materials for teaching and research purposes.

Analyzing risk factors of tuberculosis using type-2 interval-valued trapezoidal fuzzy numbers with Einstein aggregation operators extended to MCDM

The principal motive of this work is to evolve and initiate an extension from interval-valued fuzzy sets to type-2 interval-valued fuzzy sets (T2IVFS) related to weighted aggregation functions containing the Einstein operator. The chief reason for this extension is that the constancy of the terms can also be taken into data during the aggregation operation. The main goal of this article is to compose the aggregation operators and their characteristics such as the Type-2 interval-valued fuzzy Einstein weighted arithmetic aggregating operator (T2IVFEWA), Type-2 interval-valued fuzzy Einstein weighted geometric aggregating operator (T2IVFEWG), and the characteristics are expressed. At last, to intimate the effectiveness of the suggested approach and explicate the purpose of these operators, a hybrid multi-criteria decision-making problem (MCDM) to select the best risk factor for Tuberculosis (TB) is considered and the result is compared with the outcome of the existing operators and methods. Additionally, a sensitivity analysis was conducted to verify the robustness of the proposed decision-making process.

Groundwater health risk assessment and its temporal and spatial evolution based on trapezoidal fuzzy number-Monte Carlo stochastic simulation: A case study in western Jilin province

The United States Environmental Protection Agency (USEPA) Four-step-Method (FSM) is a straightforward and extensively utilized tool for evaluating regional health risks, However, the complex and heterogeneous groundwater environment system causes great uncertainty in the assessment process. Triangular stochastic simulation (TSS) possesses certain advantages in solving uncertainty problems, but its inadequacy with discrete data reveals limitations in this aspect. To solve the above problems, this study proposes to construct trapezoidal fuzzy number-Monte Carlo stochastic simulation (TFN-MCSS) to compensate for the shortcomings of the first two methods. This method adopted trapezoidal fuzzy number (TFN) analysis to comprehensively consider the characteristics of a large dispersion of water quality monitoring data and the uncertainty of the human health risk assessment (HHRA) process. Concurrently, to overcome the subjectivity and uncertainty of artificially determining the interval of TFN in traditional methods, the slope was used to select the most probable interval value (TMPIV) of TFN combined with the α-truncated set technique (α-TST) and MCSS. Based on these, a TFN-MCSS was constructed and applied to groundwater HHRA in western Jilin Province. First, the groundwater chemical characteristic determination and water quality evaluation in western Jilin were performed to identify the main pollution indicators, and the health risk effects of pollutants in groundwater of different aquifers at different time periods on adults and children were evaluated using the TFN-MCSS. The uncertainty and sensitivity were analyzed, and the primary risk control indicators were identified and compared to FSM and TSS. The results reveal that TFN-MCSS was more sensitive to data and could reduce the uncertainty of assessment process. It indicated that over a 10-year period, the health risks associated with unconfined groundwater (UW) and confined water (CW) decreased by greater than 52 %. However, the highest total non-carcinogenic risk index (THI) was 1.3-fold higher than the safety threshold, and this posed a health risk.